Find . F.dr when F(x,y) = (2x,3y²) and C is only the first quadrant part of a circle of radius 3 centered at the origin oriented clockwise.

Q: Find, correct to four decimal places, the length of the curve of intersection of the cylinder 9x2 +…

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Q: Find the equation of tangent to the circle x + y = 25 at the point (3, –4) 25 O a. y = 7x 4 13 25 O…

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Q: Find the area of the surface pf revolution geneeated by revolvinh about y= -2 the arc of the…

A: Given, So, x=y212=gy ⇒g'y=y6 limit of y is from 0 to 6 The formula for finding the area is,…

Q: Sketch the section of the surface at x = 0. y? – 42? = 4

A: Given

Q: The tangent to the circle x? + y? +3 x+ 3 y = 38 at the point (2,4) is Select one: O 11 y + 7x = 18…

A: According to the equation, we need to calculate the equation of tangent to the circle at point…

Q: Identify and sketch the graph of the following SURFACES. x² + y² = sin²z, z = [0, π]

A: Any point on the 3D plane is represented using the ordered pair: x, y, z. A graphing utility can be…

Q: * Find the equation of the tangent plane to the surface 9a2²-12xy = 5x2+5 at the ivon Ni Smob point…

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Q: Use the previous exercise to write the equation of the circle ofradius 5 and center (3, 4) in the…

A: Standard form of a equation of a circle whose center is (h, k) and radius is r is, x-h2+y-k2=r2

Q: Evaluate x+y JF dx dy x² + y² + a² over the portion of the first quadrant lying inside the circle x²…

A: Answer is given using polar coordinates.

Q: Which of the following is the equation of the tangent plane to the surface z = 3x2 +x-y²+y at the…

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Q: Find the point on the surface z = xy + 1 nearest the origin.

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Q: If C'is the circle of radius 3 centered at the origin, starting at the point (3,0) and moving around…

A: Answer and explanation is given below..

Q: Find an equation of the tangent plane to the surface z = læ? + 2y² + 2x + 1y +3 at the point (2, 4,…

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Q: Evaluate (3y - esinz)da + (7x+ V +1)dy where C is the unit circle

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Q: If the infinite curve y = e-6x, x > 0, is rotated about the x-axis, find the area of the resulting…

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Q: Find an equation of the tangent plane to the surface z = 2a2 + 1y – la – 3y – 1 at the point (5, 3,…

A: Equation of Tangent Plane to the surface F(x, y, z) = 0 at point (x0, y0, z0) is given by ∇F(x0, y0,…

Q: Evaluate |2*(x +7) – 2y ds C where C is the circle of radius 1 centered at (-3,0) and lying on the…

A: Since the integral ins on the plane x=-3 so the integral becomes:  ∫Cz3x+7-2yds=∫Cz3-3+7-2yds…

Q: Find a parametrization of the osculating circle for the parabola y = x2 when x = 1

A: It is given that, y = x2.The parametrization is, r (t) = <t, t2>That implies,

Q: Find an equation of the tangent plane to the surface at the given point. h(x, y) = In Vx? + y², (6,…

A: here h(x, y)=z then We can represent the function in the phoneF(x,y,z)=0 where F(x,y,z)=z−ln(x2+y2)…

Q: Find F. dr where C is a circle of radius 2 in the plane. x+y+z 9, centered at (4, 3, 2) and oriented…

A: Consider the given equation: And the plane x+y+z=9 centered at 4,3,2.

Q: Sketch the section of the surface at y= 0. 4. + 9 36

A: Given the equation4x2+9z2=36

Q: 3. z = (c

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Q: The tangent to the circle x? + y? +3x+3 y = 38 at the point (2 ,4) is Select one: O11 y- 7x = 58 011…

A: Equation of the circle is : x2 + y2 + 3x + 3y = 38 at point (2,4)

Q: Find the area of the surface generated when the given curve is revolved about the y-axis. y = (x^2…

A: Solution:

Q: Evaluate  ∫C (3y + exsinx) dx + (8x - ln(y3+2)) dy, where C is the circle x2 + y2 = 9  with positive…

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Q: Find, correct to four decimal places, the length of the curve of intersection of the cylinder 9x2 +…

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Q: Demonstrate that the circles (x – 2)? + y² = 2 and x? + (y – 2)2 = 2 share tangent at point (1, 1),…

A: Here we will apply the concept of finding the equation of tangent to the two circles at point (1,1).

Q: Evaluate ∫C x ds, where C is the parabolic curve x = t, y = t2, from (0, 0) to (2, 4).

A: Given: Parabolic curve x=t, y=t2 from (0,0) to (2,4)

Q: Verify Green's theorem for $[xy² dx + x²y dy], where y is the circle x? + y² = 4.

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Q: BE f : C+ C GIVEN BY AND C IS THE CIRCLE OF CENTER IN z0 = a + br AND RADIUS r f(z) = z = + yr b 20…

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Q: Find the tangent plane to the surface z = 9 – x* – y? at the point (1, -3, –1).

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Q: Find the center and radius of circle x“ + y° - 2x-4y+1=0. Prove thot (0

A: I am solving question 1. if you want other question to be solved then repost.

Q: Find the equation of tangent to the circle x + y = 25 at the point (3, –4) %3D 25 ix + O b. y = 7 O…

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Q: Prove that the curve given by the equations x=R(cost+tsint),y=R(sint−tcost) is the involute of the…

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Q: Find the area of the surface formed by rotating the curve with the equation x ^ (2) - (y ^ (2)) /…

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Q: Find the length of the tangent to the circle x² + y² − 2x − 3y − 1 = 0 from the point (2, 5)

A: Solution- The equation of the circle x2+y2-2x-3y-1=0 The point =2,5 length of the tangent

Q: Evaluate fc (2 +x 2y ) ds, where is the upper half of the unit circle x2 y2 =1

A: We have to evaluate ∮C2+x2ydS   where C is the upper half of the unit circle x2+y2=1

Q: 4. Evaluate where C is a circle of radius 2, parallel to the r - z plane and around the y-axis with…

A: Using the parametric equation of the curve C.

Q: 2. A CIRCLE IS TANGENT TO THE CINCLES x7 y? = 4 x*+ y? = 12Y+ 64 2 F IND THE LOcuS OF IT3 CENTEN AND…

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Q: Find an equation of the tangent plane to the given surface at the specified poin 8x2 + y2 - 7y, (1,…

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Q: 8. PT is tangent to circle O at P If mNP = 90°, and mMXP : 186°, find %3D %3D X/ d. mZ4 e. mZ5 f.…

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Q: -y° dr + x* dy where C is the union of the circles of radii 1 and 2 (centered at the origin).

A: Let I=∮C−y3dx+x3dy According to green theorem ∮CLdx+Mdy=∬D∂M∂x−∂L∂ydxdy Where, D is the region…

Q: dz b) C: z|= 4 is a circle calculate the integral .(z-1)(z² +9)

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Q: at (2,-1,12). Find an equation for the tangent plane to the surface z=x -2xy +4y²

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Q: If C1 is the segment connecting (x1, Y1) to (x2, Y2), find | x dy – y dx in terms of the four…

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Q: On the surface z =, in what direction is z increasing most rapidly when x =1 and y = 2.

A: A function z=f(x,y) increases most rapidly, at a point (a,b) , in the direction of the gradient…

Q: 3. y = Vx,0< x<1 ; у-ахis

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Q: 2. Write the equation of the circle with Center (-1, – 8) and Radius : V7 x2 + 2x + y? + 16y + 58 =…

A: Topic-analytic geometry

Q: $ ydx – xdy where C is the circle of radius 3 with center at (13, 28)

A: We use Green's theorem to evaluate the given line integral

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.